Finite Group Databasebeta

← #27140704

Subgroups — #27140704

Order 12416.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261316C211yesmaximal, characteristic
2#20260874C212yesmaximal, complemented
3#20260468C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#395×C97C211yesmaximal, characteristic, complemented
6C2×#13095511—21no—
7C2×#13095498—22no—
8#15690790—22no—
9#15690785—22no—
10#15690784—21no—
11#15690784C2212yes—
12#15690734—22no—
13C22⋊#11715895C2212yes—
14C22×Q8×C97C2211yescharacteristic
15C2×C4⋊C4×C97—21no—
16C2×C4⋊C4×C97C2211yescharacteristic
17#95×C97—21no—
18#95×C97C2211yescharacteristic
19#13095511—42no—
20#13095511—42no—
21C2×Dic194—21no—
22C2×D388—21no—
23C2×D388—21no—
24C2×D388D412yes—
25C2×C22⋊D97—22no—
26C2×C4×D97—22no—
27#13095498—44no—
28C2×#11715895—22no—
29C4⋊Dic97—21no—
30C4⋊Dic97D412yes—
31#13095488—22no—
32#13095488—22no—
33C2×Q8×C97—22no—
34C2×Q8×C97—42no—
35C22×C4×C97—21no—
36C22×C4×C97—21no—
37C22×C4×C97D411yescharacteristic
38C22×C4×C97D411yescharacteristic
39C22×C4×C97C2311yescharacteristic
40C4⋊C4×C97—22no—
41C4⋊C4×C97—42no—
42C23×C97C2×D411yescharacteristic
43C22×D97—22no—
44Q8×C97—24no—
45Q8×C97—42no—
46C2×C4×C97—21no—
47C2×C4×C97—21no—
48C2×C4×C97—22no—
49C2×C4×C97—22no—
50C2×C4×C97—21no—
51C2×C4×C97—41no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97SD1612yescomplemented
58C2×C4×C97C2×D412yes—
59C22⋊D97—42no—
60C22⋊D97—42no—
61C2×Dic97—22no—
62D388—42no—
63D388—42no—
64D388—81no—
65C4×D97—44no—
66Dic194—42no—
67#11715895—42no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97#12011yescharacteristic
71C22×C97C2×SD1611yescharacteristic
72C22×C97AGL(1,ℤ/8)11yescharacteristic
73C2×D97—42no—
74C2×D97—42no—
75C2×D97—42no—
76C4×C97—22no—
77C4×C97—22no—
78C4×C97—42no—
79C4×C97—41no—
80C4×C97—42no—
81Dic97—42no—
82C2×C97—21no—
83C2×C97—21no—
84C2×C97#45011yescharacteristic
85C2×C97#45611yescharacteristic
86C2×C97B(4,𝔽2)11yescharacteristic
87D97—82no—
88#1604—971nomaximal, Sylow
89C97#160411yescharacteristic, complemented, Sylow
90#537—971no—
91#448—972no—
92#395D9711yescharacteristic, complemented
93#327—972no—
94#324—971no—
95C2×#41—1941no—
96C22×Q8C2×D9711yescharacteristic
97C2×SD16—1942no—
98C22⋊Q8—1942no—
99#121—972no—
100#121—1942no—
101#121—1941no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C2×D9711yescharacteristic
104#103—1942no—
105C22⋊C8—972no—
106#95—21no—
107#95C2×D9711yescharacteristic
108#41—3882no—
109#41—3882no—
110C2×Q8—22no—
111C2×Q8—42no—
112C2×Q8—1941no—
113C2×D4—972no—
114C2×D4—1941no—
115C2×D4—1941no—
116C2×D4—1942no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—1942no—
120C22×C4C22⋊D9711yescharacteristic
121C22×C4C22⋊D9711yescharacteristic
122C22×C4C22×D9711yescharacteristic
123SD16—3884no—
124C2×C8—1942no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—972no—
128C4⋊C4—1941no—
129C22⋊C4—1942no—
130C22⋊C4—1942no—
131C23—1942no—
132C23C2×C22⋊D9711yescharacteristic
133Q8—24no—
134Q8—42no—
135Q8—3882no—
136D4—7761no—
137D4—3882no—
138D4—3882no—
139D4—3882no—
140D4—3882no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—3884no—
153C2×C4—1942no—
154C2×C4#1309549812yescomplemented
155C2×C4D4×D9712yes—
156C8—3882no—
157C22—21no—
158C22—21no—
159C22—3882no—
160C22—3882no—
161C22—3882no—
162C22#1569077811yescharacteristic
163C22C2×#1309549811yescharacteristic
164C22#1569083511yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—42no—
169C4—41no—
170C4—3882no—
171C2—21no—
172C2—21no—
173C2—7762no—
174C2#2026088511yescharacteristic
175C2#2026091711yescharacteristic
176C2#2026092111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#395×C97C2
socleC22×C97#120
prime radical#20261316C2
hypercenter#395D97
hypocenterC97#1604
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20260874C2semidirect
3. #20260468C2semidirect
4. #20260461C2semidirect
5. #395×C97C2semidirect
57. C2×C4×C97SD16semidirect
89. C97#1604semidirect
92. #395D97semidirect
154. C2×C4#13095498semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202613162
#202608742
#202604682
#202604612
#395×C972
#160497